Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-4x+4y &= -7 \\ 5x-6y &= 7\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $5x = 6y+7$ Divide both sides by $5$ to isolate $x$ $x = {\dfrac{6}{5}y + \dfrac{7}{5}}$ Substitute this expression for $x$ in the first equation. $-4({\dfrac{6}{5}y + \dfrac{7}{5}}) + 4y = -7$ $-\dfrac{24}{5}y - \dfrac{28}{5} + 4y = -7$ Simplify by combining terms, then solve for $y$ $-\dfrac{4}{5}y - \dfrac{28}{5} = -7$ $-\dfrac{4}{5}y = -\dfrac{7}{5}$ $y = \dfrac{7}{4}$ Substitute $\dfrac{7}{4}$ for $y$ in the top equation. $-4x+4( \dfrac{7}{4}) = -7$ $-4x+7 = -7$ $-4x = -14$ $x = \dfrac{7}{2}$ The solution is $\enspace x = \dfrac{7}{2}, \enspace y = \dfrac{7}{4}$.